That didn't come up too intuitive. Ok, think of two vectors in "three dimensions" (but not really). Draw X,Y,Z axes. So these pseudovectors will have a determi

That didn't come up too intuitive. Ok, think of two vectors in "three dimensions" (but not really). Draw X,Y,Z axes. So these pseudovectors will have a determinant which you can project on XY space and YZ space. Now, because our vectors have 4 dimensions and not 3, these projections will not be the same! Then your four-dimensional symplectic form (in some applications an "exterior product") is the sum of those two projections. So you're pseudovisualizing a 4D object in 3D space as you might pseudovisualize a 3D object in 2D space but with a different concept of projection/perspective/etc. because we're not in the inner product geometry, we're in the symplectic geometry.